322 lines
10 KiB
Text
322 lines
10 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: init.nat
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Authors: Floris van Doorn, Leonardo de Moura
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-/
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prelude
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import init.wf init.tactic init.num
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open eq.ops decidable
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namespace nat
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notation `ℕ` := nat
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inductive lt (a : nat) : nat → Prop :=
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| base : lt a (succ a)
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| step : Π {b}, lt a b → lt a (succ b)
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notation a < b := lt a b
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definition le [reducible] (a b : nat) : Prop := a < succ b
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notation a ≤ b := le a b
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definition pred (a : nat) : nat :=
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nat.cases_on a zero (λ a₁, a₁)
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protected definition is_inhabited [instance] : inhabited nat :=
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inhabited.mk zero
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protected definition has_decidable_eq [instance] : ∀ x y : nat, decidable (x = y)
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| has_decidable_eq zero zero := inl rfl
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| has_decidable_eq (succ x) zero := inr (λ h, nat.no_confusion h)
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| has_decidable_eq zero (succ y) := inr (λ h, nat.no_confusion h)
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| has_decidable_eq (succ x) (succ y) :=
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if H : x = y
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then inl (eq.rec_on H rfl)
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else inr (λ h : succ x = succ y, nat.no_confusion h (λ heq : x = y, absurd heq H))
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-- less-than is well-founded
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definition lt.wf [instance] : well_founded lt :=
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well_founded.intro (λn, nat.rec_on n
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(acc.intro zero (λ (y : nat) (hlt : y < zero),
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have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
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λ n₁ hlt, nat.lt.cases_on hlt
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(λ heq, nat.no_confusion heq)
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(λ b hlt heq, nat.no_confusion heq),
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aux hlt rfl))
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(λ (n : nat) (ih : acc lt n),
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acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
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have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
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λ n₁ hlt, nat.lt.cases_on hlt
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(λ (heq : succ n = succ m),
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nat.no_confusion heq (λ (e : n = m),
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eq.rec_on e ih))
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(λ b (hlt : m < b) (heq : succ n = succ b),
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nat.no_confusion heq (λ (e : n = b),
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acc.inv (eq.rec_on e ih) hlt)),
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aux hlt rfl)))
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definition measure {A : Type} (f : A → nat) : A → A → Prop :=
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inv_image lt f
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definition measure.wf {A : Type} (f : A → nat) : well_founded (measure f) :=
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inv_image.wf f lt.wf
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definition not_lt_zero (a : nat) : ¬ a < zero :=
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have aux : ∀ {b}, a < b → b = zero → false, from
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λ b H, lt.cases_on H
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(λ heq, nat.no_confusion heq)
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(λ b h₁ heq, nat.no_confusion heq),
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λ H, aux H rfl
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definition zero_lt_succ (a : nat) : zero < succ a :=
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nat.rec_on a
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(lt.base zero)
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(λ a (hlt : zero < succ a), lt.step hlt)
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definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
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have aux : ∀ {d}, d < c → b = d → a < b → a < c, from
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(λ d H, lt.rec_on H
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(λ h₁ h₂, lt.step (eq.rec_on h₁ h₂))
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(λ b hl ih h₁ h₂, lt.step (ih h₁ h₂))),
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aux H₂ rfl H₁
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definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
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lt.rec_on H
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(lt.base (succ a))
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(λ b hlt ih, lt.trans ih (lt.base (succ b)))
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definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
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have aux : ∀ {a₁}, a₁ < b → succ a = a₁ → a < b, from
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λ a₁ H, lt.rec_on H
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(λ e₁, eq.rec_on e₁ (lt.step (lt.base a)))
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(λ d hlt ih e₁, lt.step (ih e₁)),
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aux H rfl
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definition lt_of_succ_lt_succ {a b : nat} (H : succ a < succ b) : a < b :=
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have aux : pred (succ a) < pred (succ b), from
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lt.rec_on H
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(lt.base a)
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(λ (b : nat) (hlt : succ a < b) ih,
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show pred (succ a) < pred (succ b), from
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lt_of_succ_lt hlt),
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aux
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definition decidable_lt [instance] : decidable_rel lt :=
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λ a b, nat.rec_on b
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(λ (a : nat), inr (not_lt_zero a))
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(λ (b₁ : nat) (ih : ∀ a, decidable (a < b₁)) (a : nat), nat.cases_on a
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(inl !zero_lt_succ)
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(λ a, decidable.rec_on (ih a)
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(λ h_pos : a < b₁, inl (succ_lt_succ h_pos))
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(λ h_neg : ¬ a < b₁,
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have aux : ¬ succ a < succ b₁, from
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λ h : succ a < succ b₁, h_neg (lt_of_succ_lt_succ h),
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inr aux)))
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a
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definition le.refl (a : nat) : a ≤ a :=
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lt.base a
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definition le_of_lt {a b : nat} (H : a < b) : a ≤ b :=
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lt.step H
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definition eq_or_lt_of_le {a b : nat} (H : a ≤ b) : a = b ∨ a < b :=
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begin
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cases H with [b, hlt],
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apply (or.inl rfl),
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apply (or.inr hlt)
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end
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definition le_of_eq_or_lt {a b : nat} (H : a = b ∨ a < b) : a ≤ b :=
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or.rec_on H
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(λ hl, eq.rec_on hl !le.refl)
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(λ hr, le_of_lt hr)
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definition decidable_le [instance] : decidable_rel le :=
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λ a b, decidable_of_decidable_of_iff _ (iff.intro le_of_eq_or_lt eq_or_lt_of_le)
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definition le.rec_on {a : nat} {P : nat → Prop} {b : nat} (H : a ≤ b) (H₁ : P a) (H₂ : ∀ b, a < b → P b) : P b :=
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begin
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cases H with [b', hlt],
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apply H₁,
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apply (H₂ b' hlt)
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end
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definition lt.irrefl (a : nat) : ¬ a < a :=
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nat.rec_on a
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!not_lt_zero
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(λ (a : nat) (ih : ¬ a < a) (h : succ a < succ a),
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ih (lt_of_succ_lt_succ h))
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definition lt.asymm {a b : nat} (H : a < b) : ¬ b < a :=
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lt.rec_on H
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(λ h : succ a < a, !lt.irrefl (lt_of_succ_lt h))
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(λ b hlt (ih : ¬ b < a) (h : succ b < a), ih (lt_of_succ_lt h))
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definition lt.trichotomy (a b : nat) : a < b ∨ a = b ∨ b < a :=
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nat.rec_on b
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(λa, nat.cases_on a
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(or.inr (or.inl rfl))
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(λ a₁, or.inr (or.inr !zero_lt_succ)))
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(λ b₁ (ih : ∀a, a < b₁ ∨ a = b₁ ∨ b₁ < a) (a : nat), nat.cases_on a
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(or.inl !zero_lt_succ)
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(λ a, or.rec_on (ih a)
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(λ h : a < b₁, or.inl (succ_lt_succ h))
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(λ h, or.rec_on h
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(λ h : a = b₁, or.inr (or.inl (eq.rec_on h rfl)))
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(λ h : b₁ < a, or.inr (or.inr (succ_lt_succ h))))))
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a
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definition eq_or_lt_of_not_lt {a b : nat} (hnlt : ¬ a < b) : a = b ∨ b < a :=
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or.rec_on (lt.trichotomy a b)
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(λ hlt, absurd hlt hnlt)
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(λ h, h)
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definition lt_succ_of_le {a b : nat} (h : a ≤ b) : a < succ b :=
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h
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definition lt_of_succ_le {a b : nat} (h : succ a ≤ b) : a < b :=
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lt_of_succ_lt_succ h
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definition le_succ_of_le {a b : nat} (h : a ≤ b) : a ≤ succ b :=
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lt.step h
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definition succ_le_of_lt {a b : nat} (h : a < b) : succ a ≤ b :=
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succ_lt_succ h
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definition le.trans {a b c : nat} (h₁ : a ≤ b) (h₂ : b ≤ c) : a ≤ c :=
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begin
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cases h₁ with [b', hlt],
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apply h₂,
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apply (lt.trans hlt h₂)
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end
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definition lt_of_le_of_lt {a b c : nat} (h₁ : a ≤ b) (h₂ : b < c) : a < c :=
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begin
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cases h₁ with [b', hlt],
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apply h₂,
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apply (lt.trans hlt h₂)
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end
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definition lt_of_lt_of_le {a b c : nat} (h₁ : a < b) (h₂ : b ≤ c) : a < c :=
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begin
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cases h₁ with [b', hlt],
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apply (lt_of_succ_lt_succ h₂),
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apply (lt.trans hlt (lt_of_succ_lt_succ h₂))
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end
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calc_trans lt.trans
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calc_trans lt_of_le_of_lt
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calc_trans lt_of_lt_of_le
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calc_trans le.trans
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definition max (a b : nat) : nat :=
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if a < b then b else a
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definition min (a b : nat) : nat :=
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if a < b then a else b
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definition max_self (a : nat) : max a a = a :=
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eq.rec_on !if_t_t rfl
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definition max_eq_right {a b : nat} (H : a < b) : max a b = b :=
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if_pos H
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definition max_eq_left {a b : nat} (H : ¬ a < b) : max a b = a :=
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if_neg H
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definition eq_max_right {a b : nat} (H : a < b) : b = max a b :=
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eq.rec_on (max_eq_right H) rfl
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definition eq_max_left {a b : nat} (H : ¬ a < b) : a = max a b :=
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eq.rec_on (max_eq_left H) rfl
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definition le_max_left (a b : nat) : a ≤ max a b :=
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by_cases
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(λ h : a < b, le_of_lt (eq.rec_on (eq_max_right h) h))
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(λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
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definition le_max_right (a b : nat) : b ≤ max a b :=
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by_cases
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(λ h : a < b, eq.rec_on (eq_max_right h) !le.refl)
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(λ h : ¬ a < b, or.rec_on (eq_or_lt_of_not_lt h)
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(λ heq, eq.rec_on heq (eq.rec_on (eq.symm (max_self a)) !le.refl))
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(λ h : b < a,
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have aux : a = max a b, from eq_max_left (lt.asymm h),
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eq.rec_on aux (le_of_lt h)))
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definition gt [reducible] a b := lt b a
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definition decidable_gt [instance] : decidable_rel gt :=
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_
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notation a > b := gt a b
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definition ge [reducible] a b := le b a
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definition decidable_ge [instance] : decidable_rel ge :=
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_
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notation a ≥ b := ge a b
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-- add is defined in init.num
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definition sub (a b : nat) : nat :=
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nat.rec_on b a (λ b₁ r, pred r)
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notation a - b := sub a b
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definition mul (a b : nat) : nat :=
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nat.rec_on b zero (λ b₁ r, r + a)
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notation a * b := mul a b
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section
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local attribute sub [reducible]
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definition succ_sub_succ_eq_sub (a b : nat) : succ a - succ b = a - b :=
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nat.induction_on b
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rfl
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(λ b₁ (ih : succ a - succ b₁ = a - b₁),
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eq.rec_on ih (eq.refl (pred (succ a - succ b₁))))
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end
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definition sub_eq_succ_sub_succ (a b : nat) : a - b = succ a - succ b :=
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eq.rec_on (succ_sub_succ_eq_sub a b) rfl
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definition zero_sub_eq_zero (a : nat) : zero - a = zero :=
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nat.induction_on a
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rfl
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(λ a₁ (ih : zero - a₁ = zero),
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eq.rec_on ih (eq.refl (pred (zero - a₁))))
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definition zero_eq_zero_sub (a : nat) : zero = zero - a :=
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eq.rec_on (zero_sub_eq_zero a) rfl
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definition sub_lt {a b : nat} : zero < a → zero < b → a - b < a :=
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have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
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λa h₁, lt.rec_on h₁
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(λb h₂, lt.cases_on h₂
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(lt.base zero)
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(λ b₁ bpos,
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eq.rec_on (sub_eq_succ_sub_succ zero b₁)
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(eq.rec_on (zero_eq_zero_sub b₁) (lt.base zero))))
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(λa₁ apos ih b h₂, lt.cases_on h₂
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(lt.base a₁)
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(λ b₁ bpos,
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eq.rec_on (sub_eq_succ_sub_succ a₁ b₁)
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(lt.trans (@ih b₁ bpos) (lt.base a₁)))),
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λ h₁ h₂, aux h₁ h₂
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definition pred_le (a : nat) : pred a ≤ a :=
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nat.cases_on a
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(le.refl zero)
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(λ a₁, le_of_lt (lt.base a₁))
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definition sub_le (a b : nat) : a - b ≤ a :=
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nat.induction_on b
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(le.refl a)
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(λ b₁ ih, le.trans !pred_le ih)
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end nat
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